Gaussian Fluctuations for Interacting Particle Systems with Singular Kernels

We consider the asymptotic behaviour of the fluctuations for the empirical measures of interacting particle systems with singular kernels. We prove that the sequence of fluctuation processes converges in distribution to a generalized Ornstein–Uhlenbeck process. Our result considerably extends classical results to singular kernels, including the Biot–Savart law. The result applies to the point vortex model approximating the 2D incompressible Navier–Stokes equation and the 2D Euler equation. We also obtain Gaussianity and optimal regularity of the limiting Ornstein–Uhlenbeck process. The method relies on the martingale approach and the Donsker–Varadhan variational formula, which transfers the uniform estimate to some exponential integrals. Estimation of those exponential integrals follows by cancellations and combinatorics techniques and is of the type of the large deviation principle.